STATISTICS
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Probability concerns the possible outcomes
(results) of experiments
Sample space is the group of all possible outcomes
Statistics are the basis of an analysis of a sample of
information gathered about an operation.
A Sample is information gathered on a part of the
operation
Decisions are made based on that analysis.
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If a coin is tossed, the sample space contains
two possible outcomes, heads (H) or tails (T).
◦ Written {H,T}
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If a die is rolled, the sample space of the
number of dots on the upper face
◦ Written {1,2,3,4,5,6}
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If two coins are tossed, the sample space has
four possible outcomes
◦ Written {HH, HT, TH, TT}
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Probability P of an event E occurring is:
P E  
n
s
Where n = number of occurrences and s = all possible
outcomes
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Probability P of an event E not occurring E’ is:
sn
P E '  
s
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If the probability P that something will happen then 1-P is
the probability it will not happen.
P(E′) = 1 – P
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Find the probability that a 4 will result when one die
is rolled.
n = 1 and s = 6
1
P 4  
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6
Ans
Find the probability P of at least one tail when two
coins are tossed
n = 3 {HT, TH, TT} ways to get a T
s = 4 {HT, TH, TT, HH}
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P at least one tail   Ans
4
5
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Events are independent if the probability
that the second event will occur is not
affected by what happens to the first
event.
If A and B are independent events then the
probability that both A and B will occur is
P(A and
P A and B
P AB) P=BP(A)×P(B)
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A bag contains 3 yellow marbles and 4
blue marbles. A marble is drawn, replaced
and another drawn.
.Find the probability that first one is yellow
and the second one is blue.
P yellow then blue   P yellow   Pblue 
3 4 12
P  yellow then blue    
Ans
7 7 49
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sum of numbers
Mean (average) =
number of numbers
Median is the middle number of a group
that is arranged in order of size.
Mode is the value that has the greatest
frequency.
Bimodal means there are two greatest
values of equal frequency
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Find the mean, median and mode: 40 37 37 65 22
80 72
40  37  37  65  22  80  72
Mean 
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316
 45.14 Ans
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Median = 22 37 37 40 65 72 80 = 40 the middle
number
Mode is 37, number with the greatest frequency
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Quartiles (Q1, Q2 (median), Q3) divide the
items in a set of numbers into four equally
sized parts.
Arrange numbers in order from lowest to
highest.
◦ Q1 is the median of the lower half.
◦ Q2 is the median.
◦ Q3 is the median of the upper half.
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Percentiles are numbers that divide the data
into 100 equal parts
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Given: 1 2 2 4 5 5 6 7 7 8 9 10 11 13 13
14 15 15 16 18 20 20 21 25 25
Find the 60th percentile.
There are 25 numbers, so the 60th
percentile or
P60  0.6  25  15th number
P60  13 Ans
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A frequency distribution is an arrangement
of a large group of numbers where most
values are repeated
◦ One line contains a list of possible values and a
second line contains the number of times each
value was observed in a particular time
◦ The values in the first line are divided into
intervals and the data arranged in lists.
Hours
211-215 216-220 221-225 226-230 231-235
Frequency 4
9
15
22
16
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A histogram is a bar graph whose bars touch
each other
A frequency distribution can be graphed as
a histogram
◦ Use the intervals on the horizontal axis
◦ Use the frequencies on the vertical axis
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Draw a histogram of the frequency
distribution below
Hours
211-215 216-220 221-225 226-230 231-235
Frequency 4
9
15
22
16
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Draw a histogram of the frequency
distribution below
Hours
211-215 216-220 221-225 226-230 231-235
Frequency 4
9
15
22
16
25
20
15
10
5
211-215
216-220
221-225
226-230
231-235
Hours
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Range is the distance between the lowest
and highest number in in a sample.
Variance is used mainly to find the
standard deviation because it is not in the
same unit of measure as the original data.
Standard Deviation gives a measure of how
much the numbers are spread out from the
mean.
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 
n  sum of x  sum of x 
Variance 
nn  1
2
2
Where x is a measurement and n is the total
number of measurements.
Standard deviation  variance
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Find the variance and standard deviation for
the following set of numbers: 2.5, 4.6, 3.2,
5.1, 2.1, 7.3, 4.9
Variance = 7145.37  882.09
77  1
Standard deviation =
3.226
= 3.226 Ans
=1.796 Ans
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Find the probability in the following
problems.
1. Getting a “head” when a coin is tossed.
2. Drawing a blue marble from a bag containing 3
blue and 5 yellow.
3. Rolling a sum of 5 or less on a pair of die.
4. Not drawing a queen from a deck of cards.
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5. Determine the mean, median and mode of the
following set of numbers:
12,15,42,37,14,9,25,32,32,30
6. Determine the variance and the standard
deviation of the following set of numbers:
76,55,77,72,39,46,47,61,59,74,43
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1.
2.
3.
4.
5.
6.
1/2
3/8
11/36
12/13
mean = 24.7 median = 27.5 mode = 32
variance = 199.6 standard deviation =
14.13
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